Bihomomorphism

Definition

Definition with symbols

Let $G,H,K$ be groups. A map $f:G\times H\to K$ is termed a bihomomorphism if for every $g$ in $G$, the induced map $h\mapsto f(g,h)$ is a homomorphism from $H$ to $K$, and for every $h\in H$, the induced map $g\mapsto f(g,h)$ is a homomorphism from $G$ to $K$.

Bihomomorphism is a group-theoretic variant of the notion of bilinear map in vector spaces.

Facts

• Subgroup generated by image of bihomomorphism is abelian, or in other words, all the elements that can be written as images under the bihomomorphisms commute with each other.
• Kernel of a bihomomorphism implies abelian-quotient, follows from the preceding.
• Kernel of a bihomomorphism implies completely divisibility-closed